The Russian website problems.ru has a big collection of math problems. I use it a lot in my work as a math Olympiad coach. Recently I was giving a statistics lesson. While there was only one statistics problem on the website, it was a good one.

Assume that every person in every country was tested for IQ. A country’s IQ rating is the average IQ of the population. We also assume that for the duration of this puzzle no one is born and no one dies.

A group of citizens of country A emigrated to B. Show that the rating of both countries can go up.

After that a group of citizens of B (which may include former citizens of A) emigrated to A. Is it possible that the ratings of both countries go up again?

A group of citizens of A emigrated to B, and a group of citizens of B emigrated to C. As a result, the ratings of each country increased. After that the migration went the opposite way: some citizens of C moved to B, and some citizens from B moved to A. As a result, the ratings of all three counties went up once more. Is this possible? If yes, then how? If no, then why not?

7 Comments

George R.:

Very interesting Tanya! I believe that a usual statistical “common belief” -which is in fact misguided- is that such “transformations” are not possible, but in reality they are.
1st Case:I will use small sets and numbers, for clarity. A={2,3,4) av.=3 B={5,6,7} av.value=6
If we move 5 from setB to setA ,then we have: A={2,3,4,5) av.=3.5 and B={6,7} with av.=6.5
Both averages went up! The reason is quite obvious. We moved one (or some)members of a set that had av.value smaller than the av.value of the set but greater than the av.value of the new set (immigrating destination)
The reverse procedure (case 2) is also feasible.
Take 6 from set {6,7} and place it in set {2,3,4,5} .Both averages go up again.

Well, this is actually sometimes a sad story in real life. Meaning, the false interpretation of such statistics (surely no real “intelligence” got bigger ,but it would seem so, by someone who looks at “averages” and judges accordingly, without going deeper in the subject…) is common in real life situations (sometimes by sneaky persons (manipulation), or mostly just ignorants who dress up their pseudo-science with some “mathematical prestige”..)
Permit me and forgive me for posting a personal experience . My mother (in Greece) is a chronicle cancer patient, so I’ve been spotting and scrutinizing some relevant statistics for some years now. I was really surprised from the number of statistical “reports” that overlook the very same mathematical property, thus giving the wrong impression.
Let’s assume that set A is a set of “Healthy” people and set B is a set of people with Cancer.
A cancer patient at an early stage, leaves set A,thus increasing the life expectancy of set A , and enters set B (Cancer patients). But being of course (because of the early stage of her disease) “healthier” than the average of set B, her “migration” also increases the life expectancy of set B. That means, that someone who just got cancer-ill , being treated as a “statistical unit” this way, improves the “average health level” overall!!
Well, I don’t mean to say that medical or pharmaceutical statistics are mistrustful , but care and skepticism must always be there.

George R.:

Eric C.:

George R., in the first case, the problem details people moving from A to B, while your set moves numbers from B to A. Now this would be fine (just naming convention) except in the second case you again moved people from B to A, when the problem had meant to emigration to be in the opposite direction as the first case. I don’t see how this can be done using the sets you gave to increase the averages but I haven’t given more than a few minutes of thought myself.

George R.:

“The reverse procedure (case 2) is also feasible.” at my first comment is false, since I did the same migration (from B to A) again.
The reverse procedure cant’be done. So case 2 is not possible.
Case C, interestingly enough, can be done.
Let A={2,4} B={5,5,6,6,35} C={1}
Moving 2 from A to B, and the two 5’s from B to C all 3 averages increase.
REVERSE: Move 1 from set C to B, and the two sixes from B to A.